
Notice that with this choice, the separation constant \(K^2\)  is positiveThis is not so for other `standard' choices e.g. \(A= K^2 - a_f^2{\cal E}^2\) . for real \(\mathcal {E}\)  and plays the role of the total angular momentum including frame-dragging effects. Notwithstanding the separation of the radial dynamic from the polar angular one, geodesics are non-planar in general. In fact they typically give rise to quite non-trivial motion in any direction. Some simplification arises when the motion takes place in a given `hyperplane' with \(\theta = \theta _0\) . We will call these geodesics `shear-free', borrowing the terminology from the familiar case of Kerr BHs [1]}.
We will start with the two simplest cases: \(\theta =0\)  and \(\theta =\pi /2\) .
